Mechanics and Materials
Scalars, Vectors and Equilibrium
OxfordAQA International AS and A-level Physics
Scalars and vectors
- A vector is a quantity that has magnitude and direction. A scalar has magnitude only: vectors have direction and scalars do not.
- Write magnitude. Never write value for magnitude or quantity for magnitude.
- Every vector has a magnitude. Never write vectors do not have a magnitude.
- Speed is a scalar and velocity is a vector: velocity is speed in a stated direction.
- An object moving at constant speed whose direction is changing is accelerating, because velocity is a vector quantity.
| Scalar | Vector |
|---|---|
| distance | displacement |
| speed | velocity |
| mass | weight |
| time, energy, power | force, acceleration |
Adding vectors
Two vectors at right angles
resultant of two perpendicular vectorsR = √(F12 + F22)
angle of the resultant to F1tan θ = F2 / F1
- Magnitude: use Pythagoras. Direction: use tan for the angle.
- State the angle to a named direction: an angle to the horizontal or to the vertical, or mark it on the diagram.
Vectors at any angle
Method: scale drawing
- Choose a scale that makes the diagram use half the available space or more, and write the scale on the diagram.
- Draw the first vector to scale in its direction.
- Draw the next vector from the tip of the first, nose to tail, to make a triangle. Two perpendicular vectors drawn from one point make a rectangle.
- The resultant runs from the tail of the first vector to the tip of the last (the diagonal of the rectangle).
- Measure its length and use the scale to convert it to a magnitude.
- Measure its angle with a protractor, mark it on the diagram and state it to the horizontal or to the vertical.
Resolving vectors
- A force F at angle θ to a direction has a component F cos θ along that direction (adjacent to θ) and F sin θ at right angles to it (opposite θ).
- Resolve every force into two perpendicular directions, usually horizontal and vertical, and treat each direction separately.
- Balanced components in one direction: T1 cos θ1 = T2 cos θ2, each θ measured from that direction.
- A rope at angle θ to the vertical holding a weight: T cos θ equals the weight, so T is inversely proportional to cos θ. As the angle to the vertical decreases, cos θ increases and T will decrease.
On a slope
- θ is the angle of the slope to the horizontal.
- Parallel to the slope: mg sin θ. Perpendicular to the slope: the perpendicular component of weight, mg cos θ.
Equilibrium
- A body is in equilibrium when there is no resultant force and no resultant moment about any point.
- A body at rest, or travelling at constant velocity, is in equilibrium.
Vector triangle
- To find an unknown force, draw a vector triangle by the scale drawing method, with the correct directions of arrows and every angle annotated, then use the scale to obtain each unknown side.
- A right-angled triangle: use Pythagoras with a triangle of forces.
Quantities and units
| Quantity | Symbol | Unit |
|---|---|---|
| Force | F | N |
| Resultant force | R | N |
| Tension | T | N |
| Angle | θ | ° |
| Mass | m | kg |
| Gravitational field strength | g | N kg−1 |
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